n is a positive integer, and k is the product of all integer

1440 = 2*2*2*2*2*3*3*5

We have to find smallest n such that k is a multiple of 1440.

Or we have to find smallest n such that 1*2*3*4...*n is completely divisible by 1440
Or we have to find smallest n such that 1*2*3*4...*n is completely divisible by 2*2*2*2*2*3*3*5

So in the product of 1...n, we need 5 twos, 2 threes and a five.

This can be best obtained by writing the product only in prime factors.
Number - Product of prime factor - Cumulative prime factors of k
2 - 1 two - 1 two
3 - 1 three - 1 two and 1 three
4 - 2 twos - 3 twos and 1 three
5 - 1 five - 3 twos, 1 three and 1 five
6 - 1 two and 1 three - 4 twos, 2 threes and 1 five
7 - 1 seven - 4 twos, 2 threes, 1 five, and 1 seven
8 - 3 twos - 7 twos, 2 threes, 1 five and 1 seven

Thus, if n =8, k contains our required combination.

Thus answer is A.

it is a

Hey guys,

Great work here...and, honestly, I think that proving the "answer key" wrong and defending your thought process is incredibly valuable, so I'm happy for you that you had to do that. I had a question like this once for which I had to write the solution (for one of our practice tests, I think), and my boss double-checked my solution using Microsoft Excel.

Well, Excel rounds off almost all digits in huge numbers to zero, so my boss came back to me thinking I was way,way off (the number in question was something like 50!), and I had to stand my ground and prove it the way that you just did on this thread. In doing so, I got pretty confident explaining questions like this!


One quick strategic tip - if you're ever stuck on problems dealing with FACTORS or MULTIPLES, it can't really hurt to break down numbers into PRIME FACTORS like you did here. Prime factors are the building blocks of larger numbers, so it's a skill that the GMAT likes to reward as a great way to solve problems. Keep that ideology in mind and you'll have a go-to thought process for factor/multiple problems.

Oh, wow - thanks for pointing that out! I think it was indecision between:

It can really help

or

It can't really hurt

And I just picked the only combination that would give me an incorrect meaning...I just fixed that!

k=1440*a

k=1*2*..*n

Take the first smallest value (ie)8 and check whether k is multiple of 1440.

k=8*180*a

k=1.2.3.4.5.6.7.8=8*(180)*(28)

Pick A.